Dual bundle

For a vector bundle $\pi:E\to B$ with a vector space $F\simeq \R^n$ as a generic fiber, the dual bundle is a vector bundle $\pi^*:E^*\to B$ over the same base $B$ with the fiber $F^*$ dual to the fiber $F$.

The natural bilinear pairing $F\times F^*\to\R$, $(v,v^*)\mapsto\left<v,v^*\right>$ induces the natural pairing between the modules of sections $\Gamma(E)$ and $\Gamma(E^*)$ of the initial bundle and its dual, $$ \left<\cdot,\cdot\right>:\Gamma(E)\times\Gamma(E^*)\to C^\infty(B),\qquad (s,s^*)\mapsto \left< s,s^*\right>(b)=\left< s(b),s^*(b)\right>. $$